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G = D401C4order 320 = 26·5

1st semidirect product of D40 and C4 acting faithfully

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D401C4, Q162F5, D10.3D8, Dic5.17SD16, C8.3(C2×F5), C40.3(C2×C4), C40⋊C42C2, (C5×Q16)⋊1C4, C52(D82C4), (C4×D5).24D4, C52C8.15D4, C8.F53C2, C4.5(C22⋊F5), Q8.D10.2C2, C20.5(C22⋊C4), (C8×D5).11C22, C10.9(D4⋊C4), C2.10(D20⋊C4), SmallGroup(320,245)

Series: Derived Chief Lower central Upper central

C1C40 — D401C4
C1C5C10C20C4×D5C8×D5C40⋊C4 — D401C4
C5C10C20C40 — D401C4
C1C2C4C8Q16

Generators and relations for D401C4
 G = < a,b,c | a40=b2=c4=1, bab=a-1, cac-1=a3, cbc-1=a37b >

Subgroups: 346 in 58 conjugacy classes, 20 normal (all characteristic)
C1, C2, C2 [×2], C4, C4 [×3], C22 [×2], C5, C8, C8, C2×C4 [×3], D4 [×2], Q8, D5 [×2], C10, C16, C4⋊C4, C2×C8, D8, SD16, Q16, C4○D4, Dic5, C20, C20, F5, D10, D10, C4.Q8, M5(2), C4○D8, C52C8, C40, C4×D5, C4×D5, D20 [×2], C5×Q8, C2×F5, D82C4, C5⋊C16, C8×D5, D40, Q8⋊D5, C5×Q16, C4⋊F5, Q82D5, C8.F5, C40⋊C4, Q8.D10, D401C4
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], C22⋊C4, D8, SD16, F5, D4⋊C4, C2×F5, D82C4, C22⋊F5, D20⋊C4, D401C4

Character table of D401C4

 class 12A2B2C4A4B4C4D4E58A8B8C1016A16B16C16D20A20B20C40A40B
 size 111040281040404410104202020208161688
ρ111111111111111111111111    trivial
ρ21111111-1-111111-1-1-1-111111    linear of order 2
ρ3111-11-111111111-1-1-1-11-1-111    linear of order 2
ρ4111-11-11-1-11111111111-1-111    linear of order 2
ρ511-111-1-1i-i11-1-11-iii-i1-1-111    linear of order 4
ρ611-1-111-1i-i11-1-11i-i-ii11111    linear of order 4
ρ711-111-1-1-ii11-1-11i-i-ii1-1-111    linear of order 4
ρ811-1-111-1-ii11-1-11-iii-i11111    linear of order 4
ρ92220202002-2-2-220000200-2-2    orthogonal lifted from D4
ρ1022-2020-2002-22220000200-2-2    orthogonal lifted from D4
ρ112220-20-200200022-22-2-20000    orthogonal lifted from D8
ρ122220-20-20020002-22-22-20000    orthogonal lifted from D8
ρ1322-20-2020020002--2--2-2-2-20000    complex lifted from SD16
ρ1422-20-2020020002-2-2--2--2-20000    complex lifted from SD16
ρ1544004-4000-1400-10000-111-1-1    orthogonal lifted from C2×F5
ρ16440044000-1400-10000-1-1-1-1-1    orthogonal lifted from F5
ρ17440040000-1-400-10000-1-5511    orthogonal lifted from C22⋊F5
ρ18440040000-1-400-10000-15-511    orthogonal lifted from C22⋊F5
ρ194-4000000040-2-22-2-4000000000    complex lifted from D82C4
ρ204-40000000402-2-2-2-4000000000    complex lifted from D82C4
ρ218800-80000-2000-2000020000    orthogonal lifted from D20⋊C4, Schur index 2
ρ228-80000000-200020000000-1010    orthogonal faithful, Schur index 2
ρ238-80000000-20002000000010-10    orthogonal faithful, Schur index 2

Smallest permutation representation of D401C4
On 80 points
Generators in S80
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80)
(1 52)(2 51)(3 50)(4 49)(5 48)(6 47)(7 46)(8 45)(9 44)(10 43)(11 42)(12 41)(13 80)(14 79)(15 78)(16 77)(17 76)(18 75)(19 74)(20 73)(21 72)(22 71)(23 70)(24 69)(25 68)(26 67)(27 66)(28 65)(29 64)(30 63)(31 62)(32 61)(33 60)(34 59)(35 58)(36 57)(37 56)(38 55)(39 54)(40 53)
(2 28 10 4)(3 15 19 7)(5 29 37 13)(6 16)(8 30 24 22)(9 17 33 25)(11 31)(12 18 20 34)(14 32 38 40)(23 35 39 27)(26 36)(41 74 45 62)(42 61 54 65)(43 48 63 68)(44 75 72 71)(46 49 50 77)(47 76 59 80)(51 64 55 52)(53 78 73 58)(56 79 60 67)(57 66 69 70)

G:=sub<Sym(80)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80), (1,52)(2,51)(3,50)(4,49)(5,48)(6,47)(7,46)(8,45)(9,44)(10,43)(11,42)(12,41)(13,80)(14,79)(15,78)(16,77)(17,76)(18,75)(19,74)(20,73)(21,72)(22,71)(23,70)(24,69)(25,68)(26,67)(27,66)(28,65)(29,64)(30,63)(31,62)(32,61)(33,60)(34,59)(35,58)(36,57)(37,56)(38,55)(39,54)(40,53), (2,28,10,4)(3,15,19,7)(5,29,37,13)(6,16)(8,30,24,22)(9,17,33,25)(11,31)(12,18,20,34)(14,32,38,40)(23,35,39,27)(26,36)(41,74,45,62)(42,61,54,65)(43,48,63,68)(44,75,72,71)(46,49,50,77)(47,76,59,80)(51,64,55,52)(53,78,73,58)(56,79,60,67)(57,66,69,70)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80), (1,52)(2,51)(3,50)(4,49)(5,48)(6,47)(7,46)(8,45)(9,44)(10,43)(11,42)(12,41)(13,80)(14,79)(15,78)(16,77)(17,76)(18,75)(19,74)(20,73)(21,72)(22,71)(23,70)(24,69)(25,68)(26,67)(27,66)(28,65)(29,64)(30,63)(31,62)(32,61)(33,60)(34,59)(35,58)(36,57)(37,56)(38,55)(39,54)(40,53), (2,28,10,4)(3,15,19,7)(5,29,37,13)(6,16)(8,30,24,22)(9,17,33,25)(11,31)(12,18,20,34)(14,32,38,40)(23,35,39,27)(26,36)(41,74,45,62)(42,61,54,65)(43,48,63,68)(44,75,72,71)(46,49,50,77)(47,76,59,80)(51,64,55,52)(53,78,73,58)(56,79,60,67)(57,66,69,70) );

G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)], [(1,52),(2,51),(3,50),(4,49),(5,48),(6,47),(7,46),(8,45),(9,44),(10,43),(11,42),(12,41),(13,80),(14,79),(15,78),(16,77),(17,76),(18,75),(19,74),(20,73),(21,72),(22,71),(23,70),(24,69),(25,68),(26,67),(27,66),(28,65),(29,64),(30,63),(31,62),(32,61),(33,60),(34,59),(35,58),(36,57),(37,56),(38,55),(39,54),(40,53)], [(2,28,10,4),(3,15,19,7),(5,29,37,13),(6,16),(8,30,24,22),(9,17,33,25),(11,31),(12,18,20,34),(14,32,38,40),(23,35,39,27),(26,36),(41,74,45,62),(42,61,54,65),(43,48,63,68),(44,75,72,71),(46,49,50,77),(47,76,59,80),(51,64,55,52),(53,78,73,58),(56,79,60,67),(57,66,69,70)])

Matrix representation of D401C4 in GL8(𝔽241)

0012400000
00100000
2400100000
0240100000
00002221900
000022222200
00001921143895
0000131291370
,
23412411770000
117012470000
234712400000
071171240000
00001921143895
0000192114095
00002221900
000013559116176
,
0024000000
2400000000
0002400000
0240000000
00001000
0000024000
0000491270146
00002061941370

G:=sub<GL(8,GF(241))| [0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,1,1,1,1,0,0,0,0,240,0,0,0,0,0,0,0,0,0,0,0,222,222,192,13,0,0,0,0,19,222,114,129,0,0,0,0,0,0,38,137,0,0,0,0,0,0,95,0],[234,117,234,0,0,0,0,0,124,0,7,7,0,0,0,0,117,124,124,117,0,0,0,0,7,7,0,124,0,0,0,0,0,0,0,0,192,192,222,135,0,0,0,0,114,114,19,59,0,0,0,0,38,0,0,116,0,0,0,0,95,95,0,176],[0,240,0,0,0,0,0,0,0,0,0,240,0,0,0,0,240,0,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,0,1,0,49,206,0,0,0,0,0,240,127,194,0,0,0,0,0,0,0,137,0,0,0,0,0,0,146,0] >;

D401C4 in GAP, Magma, Sage, TeX

D_{40}\rtimes_1C_4
% in TeX

G:=Group("D40:1C4");
// GroupNames label

G:=SmallGroup(320,245);
// by ID

G=gap.SmallGroup(320,245);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,232,387,675,794,80,1684,851,102,6278,3156]);
// Polycyclic

G:=Group<a,b,c|a^40=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^3,c*b*c^-1=a^37*b>;
// generators/relations

Export

Character table of D401C4 in TeX

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